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Topological strong-field physics on sub-laser-cycle timescale

Nature Photonics volume 13, pages 849–854 (2019)Cite this article

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Abstract

The sub-laser-cycle timescale of the electronic response to strong fields enables attosecond dynamical imaging in atoms, molecules and solids1,2,3,4, with optical tunnelling and high-harmonic generation the hallmarks of attosecond optical spectroscopy2,5,6,7. Topological insulators are intimately linked with electron dynamics, as manifested via the chiral edge currents8, but it is unclear if and how topology leaves its mark on optical tunnelling and sub-cycle electronic response. Here, we identify distinct bulk topological effects on directionality and timing of currents arising during electron injection into conduction bands. We show that electrons tunnel differently in trivial and topological insulators, for the same band structure, and identify the key role of the Berry curvature in this process. These effects map onto topologically dependent attosecond delays and helicities of emitted harmonics that record the phase diagram of the system. Our findings create new roadmaps in studies of topological systems, building on the ubiquitous properties of the sub-laser-cycle strong-field response—a unique mark of attosecond science.

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Fig. 1: The Haldane system in two phases.
Fig. 2: Helicity of harmonics in topological insulators.
Fig. 3: Topologically dependent time delays in harmonic emission.
Fig. 4: Topological signal in injected currents.

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Data availability

The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.

References

  1. Krausz, F. & Ivanov, M. Attosecond physics. Rev. Mod. Phys. 81, 163–234 (2009).

    Article  ADS  Google Scholar 

  2. Baker, S. et al. Probing proton dynamics in molecules on an attosecond time scale. Science 312, 424–427 (2006).

    Article  ADS  Google Scholar 

  3. Shafir, D. et al. Resolving the time when an electron exits a tunnelling barrier. Nature 485, 343–346 (2012).

    Article  ADS  Google Scholar 

  4. Hohenleutner, M. et al. Real-time observation of interfering crystal electrons in high-harmonic generation. Nature 523, 572–575 (2015).

    Article  ADS  Google Scholar 

  5. Smirnova, O. et al. High harmonic interferometry of multi-electron dynamics in molecules. Nature 460, 972–977 (2009).

    Article  ADS  Google Scholar 

  6. Eckart, S. et al. Ultrafast preparation and detection of ring currents in single atoms. Nat. Phys. 14, 701–704 (2018).

    Article  Google Scholar 

  7. Ghimire, S. et al. Observation of high-order harmonic generation in a bulk crystal. Nat. Phys. 7, 138–141 (2010).

    Article  Google Scholar 

  8. Hasan, M. Z. & Kane, C. L. Colloquium: topological insulators. Rev. Mod. Phys. 82, 3045–3067 (2010).

    Article  ADS  Google Scholar 

  9. Schubert, O. et al. Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations. Nat. Photon. 8, 119–123 (2014).

    Article  ADS  Google Scholar 

  10. Luu, T. T. et al. Extreme ultraviolet high-harmonic spectroscopy of solids. Nature 521, 498–502 (2015).

    Article  ADS  Google Scholar 

  11. Vampa, G. et al. All-optical reconstruction of crystal band structure. Phys. Rev. Lett. 115, 193603 (2015).

    Article  ADS  Google Scholar 

  12. Tancogne-Dejean, N., Mücke, O. D., Kärtner, F. X. & Rubio, A. Impact of the electronic band structure in high-harmonic generation spectra of solids. Phys. Rev. Lett. 118, 087403 (2017).

    Article  ADS  Google Scholar 

  13. McDonald, C. R., Vampa, G., Corkum, P. B. & Brabec, T. Interband Bloch oscillation mechanism for high-harmonic generation in semiconductor crystals. Phys. Rev. A 92, 033845 (2015).

    Article  ADS  Google Scholar 

  14. Bauer, D. & Hansen, K. K. High-harmonic generation in solids with and without topological edge states. Phys. Rev. Lett. 120, 177401 (2018).

    Article  ADS  Google Scholar 

  15. Silva, R. E. F., Blinov, I. V., Rubtsov, A. N., Smirnova, O. & Ivanov, M. High-harmonic spectroscopy of ultrafast many-body dynamics in strongly correlated systems. Nat. Photon. 12, 266–270 (2018).

    Article  ADS  Google Scholar 

  16. Liu, H. et al. High-harmonic generation from an atomically thin semiconductor. Nat. Phys. 13, 262–265 (2017).

    Article  Google Scholar 

  17. Luu, T. T. & Wörner, H. J. Measurement of the Berry curvature of solids using high-harmonic spectroscopy. Nat. Commun. 9, 916 (2018).

    Article  ADS  Google Scholar 

  18. Thouless, D. J., Kohmoto, M., Nightingale, M. P. & den Nijs, M. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett. 49, 405–408 (1982).

    Article  ADS  Google Scholar 

  19. Reimann, J. et al. Subcycle observation of lightwave-driven Dirac currents in a topological surface band. Nature 562, 396–400 (2018).

    Article  ADS  Google Scholar 

  20. Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: condensed-matter realization of the ‘parity anomaly’. Phys. Rev. Lett. 61, 2015–2018 (1988).

    Article  ADS  MathSciNet  Google Scholar 

  21. Jotzu, G. et al. Experimental realization of the topological Haldane model with ultracold fermions. Nature 515, 237–240 (2014).

    Article  ADS  Google Scholar 

  22. Kane, C. L. & Mele, E. J. Z 2 topological order and the quantum spin Hall effect. Phys. Rev. Lett. 95, 146802 (2005).

    Article  ADS  Google Scholar 

  23. Mak, K. F., McGill, K. L., Park, J. & McEuen, P. L. The valley Hall effect in MoS2 transistors. Science 344, 1489–1492 (2014).

    Article  ADS  Google Scholar 

  24. Vampa, G. & Brabec, T. Merge of high harmonic generation from gases and solids and its implications for attosecond science. J. Phys. B 50, 083001 (2017).

    Article  ADS  Google Scholar 

  25. Ndabashimiye, G. et al. Solid-state harmonics beyond the atomic limit. Nature 534, 520–523 (2016).

    Article  ADS  Google Scholar 

  26. Hawkins, P. G., Ivanov, M. Y. & Yakovlev, V. S. Effect of multiple conduction bands on high-harmonic emission from dielectrics. Phys. Rev. A 91, 013405 (2015).

    Article  ADS  Google Scholar 

  27. Zurrón-Cifuentes, Ó., Boyero-García, R., Hernández-García, C., Picón, A. & Plaja, L. Optical anisotropy of non-perturbative high-order harmonic generation in gapless graphene. Opt. Express 27, 7776–7786 (2019).

    Article  ADS  Google Scholar 

  28. Chacón, A. et al. Observing topological phase transitions with high harmonic generation. Preprint at https://arxiv.org/abs/1807.01616 (2018).

  29. Barth, I. & Smirnova, O. Nonadiabatic tunneling in circularly polarized laser fields: physical picture and calculations. Phys. Rev. A 84, 063415 (2011).

    Article  ADS  Google Scholar 

  30. Cireasa, R. et al. Probing molecular chirality on a sub-femtosecond timescale. Nat. Phys. 11, 654–658 (2015).

    Article  Google Scholar 

  31. Chang, C.-Z. et al. Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science 340, 167–170 (2013).

    Article  ADS  Google Scholar 

  32. Sala, S., Förster, J. & Saenz, A. Ultracold-atom quantum simulator for attosecond science. Phys. Rev. A 95, 011403 (2017).

    Article  ADS  Google Scholar 

  33. Lu, L., Joannopoulos, J. D. & Soljačić, M. Topological photonics. Nat. Photon. 8, 821–829 (2014).

    Article  ADS  Google Scholar 

  34. Silva, R. E. F., Martín, F. & Ivanov, M. High harmonic generation in crystals using maximally localized Wannier functions. Preprint at https://arxiv.org/abs/1904.00283 (2019).

  35. Ashcroft, N. & Mermin, N. Solid State Physics (Cengage Learning, 2011).

  36. Keldysh, L. V. Ionization in the field of a strong electromagnetic wave. Sov. Phys. JETP 20, 1307–1314 (1965).

    MathSciNet  Google Scholar 

  37. Vampa, G. et al. Theoretical analysis of high-harmonic generation in solids. Phys. Rev. Lett. 113, 073901 (2014).

    Article  ADS  Google Scholar 

  38. Lewenstein, M., Balcou, P., Ivanov, M. Y., L’Huillier, A. & Corkum, P. B. Theory of high-harmonic generation by low-frequency laser fields. Phys. Rev. A 49, 2117–2132 (1994).

    Article  ADS  Google Scholar 

  39. Ivanov, M. Y., Spanner, M. & Smirnova, O. Anatomy of strong field ionization. J. Mod. Opt. 52, 165–184 (2005).

    Article  ADS  Google Scholar 

  40. Smirnova, O., Spanner, M. & Ivanov, M. Anatomy of strong field ionization II: to dress or not to dress? J. Mod. Opt. 54, 1019–1038 (2007).

    Article  ADS  Google Scholar 

  41. Barth, I. & Smirnova, O. Nonadiabatic tunneling in circularly polarized laser fields. II. Derivation of formulas. Phys. Rev. A 87, 013433 (2013).

    Article  ADS  Google Scholar 

  42. Pedatzur, O. et al. Attosecond tunnelling interferometry. Nat. Phys. 11, 815–819 (2015).

    Article  Google Scholar 

  43. Mairesse, Y. et al. High harmonic spectroscopy of multichannel dynamics in strong-field ionization. Phys. Rev. Lett. 104, 213601 (2010).

    Article  ADS  Google Scholar 

  44. Hartung, A. et al. Electron spin polarization in strong-field ionization of xenon atoms. Nat. Photon. 10, 526–528 (2016).

    Article  ADS  Google Scholar 

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Acknowledgements

R.E.F.S. and M.I. acknowledge support from Engineering and Physical Sciences Research Council/Defence Science and Technology Laboratory (EPSRC/DSTL) Multidisciplinary University Research Initiative (MURI) grant EP/N018680/1. R.E.F.S. acknowledges support from the European Research Council Starting Grant (ERC-2016- STG714870). Á.J.-G. and M.I. acknowledge support from the Deutsche Forschungsgemeinschaft (DFG) Quantum Dynamics in Tailored Intense Fields (QUTIF) grant IV 152/6-1. O.S. acknowledges support from the DFG Schwerpunktprogramm 1840 ‘Quantum Dynamics in Tailored Intense Fields’ project SM 292/5-1, and Molecular Electron Dynamics Investigated by Intense Fields and Attosecond Pulses (MEDEA) project, which has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 641789. B.A. received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 706538.

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Author notes

  1. These authors contributed equally: R. E. F. Silva, Á. Jiménez-Galán.

Authors and Affiliations

  1. Max-Born-Institute, Berlin, Germany

    R. E. F. Silva, Á. Jiménez-Galán, O. Smirnova & M. Ivanov

  2. Department of Theoretical Condensed Matter Physics, Universidad Autónoma de Madrid, Madrid, Spain

    R. E. F. Silva

  3. CeFEMA, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal

    B. Amorim

  4. Department of Physics, Center of Physics, and QuantaLab, University of Minho, Campus of Gualtar, Braga, Portugal

    B. Amorim

  5. Technische Universität Berlin, Ernst-Ruska-Gebäude, Berlin, Germany

    O. Smirnova

  6. Department of Physics, Humboldt University, Berlin, Germany

    M. Ivanov

  7. Blackett Laboratory, Imperial College London, South Kensington Campus, London, UK

    M. Ivanov

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Contributions

R.E.F.S., Á.J.-G. and M.I. developed the idea. R.E.F.S. developed the numerical code, Á.J.-G. performed the simulations and analysed the data. R.E.F.S., Á.J.-G. and O.S. developed the analytical analysis I. B.A., O.S. and M.I. developed the analytical analysis II. Á.J.-G. and M.I. wrote the main part of the manuscript, which was discussed by all authors.

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Correspondence to R. E. F. Silva or Á. Jiménez-Galán.

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Silva, R.E.F., Jiménez-Galán, Á., Amorim, B. et al. Topological strong-field physics on sub-laser-cycle timescale. Nat. Photonics 13, 849–854 (2019). https://doi.org/10.1038/s41566-019-0516-1

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